The present invention relates to a surface acoustic wave device using lithium tetraborate single crystal (Li.sub.2 B.sub.4 O.sub.7).
Surface acoustic wave devices are circuit elements which convert electric signals into surface waves for the signal processing, and are used in filters, resonators, delay lines, etc. Usually a surface acoustic wave device includes metal electrodes, the so-called interdigital transducers (IDTs) on a piezoelectric elastic substrate, whereby electric signals are converted into or reconverted from surface acoustic waves.
Rayleigh waves are dominantly used as the surface acoustic waves. Rayleigh waves are surface waves which propagate on elastic surfaces and propagate without radiating its energy into the piezoelectric substrate, i.e., theoretically without propagation losses. The known substrate material of surface acoustic wave devices using Rayleigh waves are quartz and lithium tantalate (LiTaO.sub.3). Quartz has good temperature stability but poor piezoelectricity. Conversely lithium tantalate has good piezoelectricity but poor temperature stability. Recently noted as a material which satisfies both properties is lithium tetraborate single crystal (refer to, e.g., Japanese Patent Publication Nos. 44169/1990 and 40044/1988).
Another alternative has been proposed, namely the use of surface acoustic waves (leaky surface acoustic waves, pseudo surface waves) which propagate by radiating a part of their energy in the direction of depth of an elastic body. Generally leaky waves are not usable in surface acoustic wave devices because of their high propagation losses due to the diffusion, but depending on specific cut angles and propagation directions, their propagation losses can be reduced usably in surface acoustic wave devices. For example, it is known that a surface acoustic wave device of lithium tantalate cut in 36.degree. Y-X-cut which can provide a propagation velocity of about 4200 m/sec.
Characteristics of Rayleigh waves and leaky waves can be simulated by the relational expressions which will be explained below (J. J. Campbell, W. R. Jones, "A Method for Estimating Optimal Crystal Cuts and Propagation Directions for Excitation of Piezoelectric Surface Waves" IEEE Trans on Sonics and Ultrasonics, Vol. SU-15, No. 4, pp. 209-217 (1968), and T. C. Lim, G. W. Farnell, "Character of Pseudo Surface Waves on Anisotropic Crystals", Journal of Acoustic Society of America, Vol. 45, No. 4, pp 845-851 (1968)).
Generally propagation characteristics of surface acoustic waves propagating on piezoelectric surfaces can be given by solving an equation of motion and a charge equation given by a Maxwell's equation under quasi-static approximation. The equation of motion and the charge equation are expressed as follows. EQU C.sub.ijkl U.sub.k.li +e.sub.kij .phi..sub..ki =.rho.U.sub.j( 1) EQU e.sub.ikl U.sub.k.li .epsilon..sub.ik .phi..sub..ki =0 (2)
where C.sub.ijkl (i, j, k, l=1, 2, 3) represents a tensor of an elastic constant; e.sub.ikl (i, k, l=1, 2, 3) represents a tensor of a piezoelectric constant; .epsilon..sup.ik (i, k=1, 2, 3) represents a tensor of a dielectric constant; and .rho. represents a density.
U.sub.i represents displacements in respective directions of the coordinate system of FIG. 1 (where X.sub.1 indicates a propagation direction of a surface acoustic wave; X.sub.2 indicates a direction perpendicular to the propagation direction X.sub.1 of the surface acoustic wave included in the elastic substrate surface; X.sub.3 indicates a direction perpendicular to the X.sub.1 and X.sub.2). .phi. represents an electrostatic potential. Displacements U.sub.i and an electrostatic potential .phi. are expressed by the following equations. EQU U.sub.i =.beta..sub.i .multidot.e.sup.jk(ax3+x1-vt) ( 3) EQU .phi.=.beta..sub.4 .multidot.e.sup.jk(ax3+x1-vt) ( 4)
where .alpha. represents decay constant in the direction X.sub.3 ; .beta..sub.i represents an amplitude constant; k represents a wave number; t represents a time; and v represents a phase velocity (propagation velocity).
First, the procedure of the Rayleigh wave simulation will be explained. The equation of motion (1) and the charge equation (2) are replaced by the equation (3) expressing displacements U.sub.i and the equation (4) expressing an electrostatic potential .phi. to be rewritten with respect to an amplitude constant .beta..sub.i. And an eighth-order equation of the decay constants a is given. Assuming that a phase velocity v is a real number, this equation gives decay constants a in a solution of a complex conjugate.
For a surface acoustic wave, an amplitude of the wave must be decayed in the direction of depth of the substrate, and for decay constants a are selected solutions having minus imaginary parts, [Im(.beta.(n))&lt;0, n=1, 2, 3, 4]. Four amplitude constants .beta..sub.1 -.beta..sub.4 are calculated corresponding to the respective selected decay constants .alpha.. By referring to the corresponding amplitude constant .beta..sub.1 -.beta..sub.4, it is found that the four selected decay constants .alpha. correspond respectively to a longitudinal wave component having a displacement in the direction x.sub.1 as the main component, two kinds of shear components having as the main components a displacement in the direction x.sub.2 or x.sub.3, and an electromagnetic component having an electrostatic potential as the main component. Based on the fact that these four components of the surface acoustic wave can propagate, displacements U.sub.i and electrostatic potentials .phi. of the surface acoustic wave in the respective directions which the surface acoustic wave can propagate are expressed in a linear combination of four modes as expressed by the following equations. ##EQU1## where A.sup.(n) represents an amplitude ratio among the respective modes.
Then boundary conditions are given to the above equations (5) and (6), whereby propagation characteristics of the surface acoustic wave are given. The mechanical boundary conditions are that the normal components of stress at surface are zero [T.sub.13 =T.sub.23 =T.sub.33 =0 at x.sub.3 =0]. For free surface (electrically open), the electrical boundary condition requires that the normal component of electric flux density at surface is zero [D.sub.3 =0 at x.sub.3 =0]. For shorted surface (electrically short), the electrical boundary condition requires that electrostatic potential is zero [.phi.=0 at x.sub.3 =0]. By giving a phase velocity v which satisfies these boundary conditions, propagation characteristics of the Rayleigh wave of the surface acoustic wave can be solved.
Next, the simulation procedure of the leaky wave will be explained. In the above-described simulation of the Rayleigh wave, when the equations (3) and (4) are substituted in the equations (1) and (2) to give decay constants a, depending an assumed value of a phase velocity v, sometimes a solution of decay constants .alpha. is not given in a complex conjugate, but in a real number. To be specific, when a phase velocity v faster than the Rayleigh wave is assumed, a part of decay constants .alpha. corresponding to one of the shear wave components (hereinafter called "the first shear wave component") are real. A component which is not decayed in the direction of depth of the piezoelectric substrate exists. Accordingly all the energy of the surface acoustic wave is not concentrated on the piezoelectric surface, and a part of the energy radiates in the direction of depth of the piezoelectric substrate. Propagation losses take place.
In this case, when the phase velocity v is simulated in a complex number as a mathematical expression of a propagation loss, the coefficients of the eighth-order equation for giving the decay constants .alpha. are in a complex number. Out of eight solutions of the decay constants .alpha., three solutions whose amplitudes decay in the direction of depth of the substrate are selected for the three components other than the first shear wave component. For the first shear wave component, another solution whose amplitude does not decay in the direction of depth of the piezoelectric substrate surface is selected. Then the boundary conditions described above in connection with the equations (5) and (6) are applied to give propagation characteristics of the leaky wave of the surface acoustic wave, which is generally called "leaky wave" ("leaky surface acoustic wave" or "pseudo surface acoustic wave").
But in the surface acoustic wave device including the substrate of lithium tetraborate, the propagation velocity of the Rayleigh wave is relatively slow, and the leaky wave, whose propagation loss is sufficiently low, is not found. Accordingly such surface acoustic wave device has found it difficult to process signals of higher frequencies.
That is, an electrode width of interdigital transducers (IDTs), and an inter-electrode gap thereof are normally set to be 1/4 of a wavelength of a surface acoustic wave corresponding to a signal frequency to be processed. The propagation velocity of Rayleigh waves on lithium tetraborate single crystal is about 3400 m/sec, and an electrode width and an inter-electrode gap of below 1 .mu.m are necessary to process signal frequencies of above 1 GHz. Accordingly fabrication yields of the interdigital transducers are lowered, and thus the fabrication of surface acoustic wave devices becomes very difficult.